In another variation of Nim, players are allowed to remove only perfect square numbers of beans and apply this condition. How many beans should you as the first player select to guarantee that you will win the game?

To determine how many beans you should select as the first player to guarantee a win, let's analyze the game of Nim with the condition that players can only remove perfect square numbers of beans.

In Nim, the key concept is to force your opponent into a losing position. A losing position is one where no matter how many beans the other player removes, you can always make a move that leaves your opponent with no winning moves.

To find the winning strategy, we can examine different scenarios and analyze the possible moves for each player.

Let's consider the number of beans available in the game:

1. If there is only one bean, then the first player cannot make a move, so they lose. In this case, there is no winning strategy.

2. If there are two beans, the first player can only remove one bean (a perfect square) and leave the opponent with one bean. The second player cannot make a move and loses. In this case, the first player wins by selecting one bean.

3. If there are three beans, the first player can remove two beans (two perfect squares) and leave the opponent with one bean. The second player cannot make a move and loses. In this case, the first player wins by selecting two beans.

Analyzing the scenarios mentioned above, we can observe that if the number of beans is a multiple of three, the first player will lose. Consequently, for any number of beans that is not divisible by three, the first player will be always able to force a win.

Therefore, to guarantee a win as the first player, you should select the number of beans that is not divisible by three.